My question is exactly my title: What are some examples of (set theoretic) forcing in model theory?
I have been studying (combinatorial) set theory and model theory (independently of one another) for some time now, and I want to know how the former can be applied to the latter. The wikipedia page on forcing states: "Forcing has also been used in model theory but it is common in model theory to define genericity directly without mention of forcing". I have, regrettably, never encountered this scenario and would like a concrete example.
Ideally, any non-trivial application of forcing to model theory would be nice.
Thanks
You should probably take a look at the classification theory text by Shelah. There area lot of results there are based on additional cardinal hypothesis, and combinatorial principles and has (set-theoretic) forcing in the background to establish relative consistency. There are places where you use forcing to establish the result also (however I can't think of a exact reference expect to say it comes up in the study of models of size $\aleph_{1}$ that are atomic but not prime) These types of results also appear in works about AECs (abstratct elementary classes).
I'm however a little confused by the article though. There is a notion of forcing based on the same ideas that is used in model theory that is called model theoretic forcing and I think that this what the article is referring to. It is due to Abraham Robinson. A good place to get details about this would be the book "Building models by Games" by W.Hodges.
EDIT: You also asked how to apply combinatorial set theory to model theory. For example most of the arguments in classification theory use a lot of set theoretic combinatorics, for example the proof that unstable theories have $2^{\kappa}$ models for any uncountable $\kappa$ comes to mind.